Global wellposedness and scattering for the focusing energy-critical nonlinear Schrodinger equations of fourth order in the radial case

Abstract

We consider the focusing energy-critical nonlinear Schrödinger equation of fourth order $i{u}_t+{\Delta }^{2}u=|u{|}^{\frac{8}{d-4}}u$. We prove that if a maximal-lifespan radial solution $u:I\times {\mathbb{R}}^d\to \mathbb{C}$ obeys $\underset{t\in I}{\sup }\Vert \Delta u(t){\Vert }_2<\Vert \Delta W{\Vert }_2$, then it is global and scatters both forward and backward in time. Here $W$ denotes the ground state, which is a stationary solution of the equation. In particular, if a solution has both energy and kinetic energy less than those of the ground state $W$ at some point in time, then the solution is global and scatters.

arXiv comment37 pages, no figure

References in corpus (10)

• Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrodinger equation in the radial case
• The mass-critical nonlinear Schrödinger equation with radial data in dimensions three and higher
• The defocusing energy-critical nonlinear Schrödinger equation in higher dimensions
• The cubic nonlinear Schrödinger equation in two dimensions with radial data
• Wellposedness of Cauchy problem for the Fourth Order Nonlinear Schrödinger Equations in Multi-dimensional Spaces
• Scattering for the non-radial 3D cubic nonlinear Schroedinger equation
• The focusing energy-critical nonlinear Schrödinger equation in dimensions five and higher
• Global well-posedness and scattering for the defocusing energy-critical nonlinear Schrödinger equation in ${\mathbb{R}}^{1+4}$
• Global well-posedness and scattering for the higher-dimensional energy-critical non-linear Schrodinger equation for radial data
• Scattering for H^1/2 bounded solutions to the cubic, defocusing NLS in 3 dimensions

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